Theorem bj-cbvex4vv

Description: Version of cbvex4v with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-cbvex4vv.1	⊢ ( ( 𝑥 = 𝑣 ∧ 𝑦 = 𝑢 ) → ( 𝜑 ↔ 𝜓 ) )
	bj-cbvex4vv.2	⊢ ( ( 𝑧 = 𝑓 ∧ 𝑤 = 𝑔 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	bj-cbvex4vv	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜑 ↔ ∃ 𝑣 ∃ 𝑢 ∃ 𝑓 ∃ 𝑔 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		bj-cbvex4vv.1	⊢ ( ( 𝑥 = 𝑣 ∧ 𝑦 = 𝑢 ) → ( 𝜑 ↔ 𝜓 ) )
2		bj-cbvex4vv.2	⊢ ( ( 𝑧 = 𝑓 ∧ 𝑤 = 𝑔 ) → ( 𝜓 ↔ 𝜒 ) )
3	1	2exbidv	⊢ ( ( 𝑥 = 𝑣 ∧ 𝑦 = 𝑢 ) → ( ∃ 𝑧 ∃ 𝑤 𝜑 ↔ ∃ 𝑧 ∃ 𝑤 𝜓 ) )
4	3	cbvex2vw	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜑 ↔ ∃ 𝑣 ∃ 𝑢 ∃ 𝑧 ∃ 𝑤 𝜓 )
5	2	cbvex2vw	⊢ ( ∃ 𝑧 ∃ 𝑤 𝜓 ↔ ∃ 𝑓 ∃ 𝑔 𝜒 )
6	5	2exbii	⊢ ( ∃ 𝑣 ∃ 𝑢 ∃ 𝑧 ∃ 𝑤 𝜓 ↔ ∃ 𝑣 ∃ 𝑢 ∃ 𝑓 ∃ 𝑔 𝜒 )
7	4 6	bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜑 ↔ ∃ 𝑣 ∃ 𝑢 ∃ 𝑓 ∃ 𝑔 𝜒 )